Amortizing Perpetual Options

TL;DR

This paper introduces amortizing perpetual options (AmPOs), a fungible variant of perpetual options that use implicit amortization to maintain fungibility and can be valued analytically through a reduction to vanilla American options.

Contribution

The paper proposes AmPOs, a novel class of options with implicit amortization, and derives analytical valuation formulas and Greeks, enabling better understanding of their behavior.

Findings

AmPOs preserve fungibility across units.

Valuation reduces to vanilla American options on dividend-paying assets.

Amortization rate influences option sensitivity and tradeoffs.

Abstract

In this work, we introduce amortizing perpetual options (AmPOs), a fungible variant of continuous-installment options suitable for exchange-based trading. Traditional installment options lapse when holders cease their payments, destroying fungibility across units of notional. AmPOs replace explicit installment payments and the need for lapsing logic with an implicit payment scheme via the decay of the claimable notional. This amortization ensures all units evolve identically, preserving fungibility. We demonstrate that AmPO valuation can be reduced to an equivalent vanilla perpetual American option on a dividend-paying asset. This enables analytical expressions for the exercise boundaries and risk-neutral valuations for calls and puts. These formulas and relations allow us to derive the Greeks and study comparative statics with respect to the amortization rate. Illustrative numerical case studies demonstrate how the amortization rate shapes option behavior and reveal the resulting tradeoffs in the effective volatility sensitivity.